Riemann surface intuition.

Question

In my complex variables notes it says that the multivalued $n$-th root function $w=z^{\frac{1}{n}}$ becomes single-valued on an appropriately constructed Riemann surface. It says how to go about producing such a surface, but it's not very detailed. I'm not really into geometry and attempting to visualise what's going on makes my head hurt. I would appreciate a relatively simple exposition of this concept, not doing into overly technical details, and explaning the intuition and motivation behind the concept and how it's applied.

Answer 1

Algebraic intuition is to note that different values correspond to dividing the argument which is equivalent up to multiples of $2\pi$ by $n$.

Define a surface parametrized by $r>0$, the magnitude , and $0<\theta<2\pi n$, the argument.

Then map $(r,\theta) \mapsto re^{i\theta/n}$ for which we can visualize the domain as $n$ copies of the complex plane cut on the positive real axis and stitched together. The first copy corresponds to $0<\theta<2\pi$, the $j$-th copy to $2\pi(j-1)<\theta<2\pi j$. Note $(r,0)=(r,2\pi n)$ on the surface and map to the same point in the complex plane.